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Abstract

A simple implementation of plane wave method is presented for modeling photonic crystals with arbitrary shaped ‘atoms’. The Fourier transform for a single ‘atom’ is first calculated either by analytical Fourier transform or numerical FFT, then the shift property is used to obtain the Fourier transform for any arbitrary supercell consisting of a finite number of ‘atoms’. To ensure accurate results, generally, two iterating processes including the plane wave iteration and grid resolution iteration must converge. Analysis shows that using analytical Fourier transform when available can improve accuracy and avoid the grid resolution iteration. It converges to the accurate results quickly using a small number of plane waves. Coordinate conversion is used to treat non-orthogonal unit cell with non-regular ‘atom’ and then is treated by standard numerical FFT. MATLAB source code for the implementation requires about less than 150 statements, and is freely available at http://www.lions.odu.edu/~sguox002.

Fig. 3. Convergence of TM mode. (a) Convergence of the first band. (b) The iteration errors for the first 10 bands. A uniform mesh with different resolution is used to represent the unit cell, and each grid is averaged by a 10×10 submesh.

Fig. 4. Convergence of TE mode. (a) Convergence of the first band. (b) The iteration errors for the first 10 bands. A uniform mesh with different resolution is used to represent the unit cell, and each grid is averaged by a 10×10 submesh.

Fig. 5. Eigen frequency convergence as a function of grid resolution for TM mode in a 2D triangular lattice. 225 plane waves are used for this calculation. Line with ‘o’: grid is averaged by a 10×10 submesh; line with ‘+’:not averaged.

Fig. 6. Eigen frequency convergence as a function of grid resolution for TE mode in a 2D triangular lattice. 225 plane waves are used for this calculation. Line with ‘o’: grid is averaged by a 10×10 submesh; line with ‘+’: not averaged.